Abstract
The theory of the minimal spanning tree (MST) of a connected graph whose edges are assigned lengths according to independent identically distributed random variables is developed from two directions. First, it is shown how the Tutte polynomial for a connected graph can be used to provide an exact formula for the length of the minimal spanning tree under the model of uniformly distributed edge lengths. Second, it is shown how the theory of local weak convergence provides a systematic approach to the asymptotic theory of the length of the MST and related power sums. Consequences of these investigations include (1) the exact rational determination of the expected length of the MST for the complete graph Kn for 2 ≤ n ≤ 9 and (2) refinements of the results of Penrose (1998) for the MST of the d-cube and results of Beveridge, Frieze, and McDiarmid (1998) and Frieze, Ruzink6, and Thoma (2000) for graphs with modest expansion properties. In most cases, the results reviewed here have not reached their final form, and they should be viewed as part of work-in-progress.
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References
D.J. Aldous (1992) Asymptotics in the random assignment problem Probab. Th. Rel. Fields, 93, 507–534.
D.J. Aldous (2001) The ç(2) limit in the random assignment problem Random Structures Algorithms, 18, 381–418.
D.J. Aldous and J.M. Steele (2002) The Objective Method and the Theory of Local Weak Convergence in Discrete and Combinatorial Probability (ed. H. Kesten), Springer-Verlag, New York. [in press]
N. Alon, A. Frieze, and D. Welsh (1994) Polynomial time randomized approximation schemes for the Tutte polynomial of dense graphs pp. 24–35 in the Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (S. Goldwasser, ed.), IEEE Computer Society Press, 1994.
F. Avram and D. Bertsimas (1992) The minimum spanning tree constant in geometric probability and under the independent model: a unified approach Annals of Applied Probability 2, 113–130.
A. Beveridge, A. Frieze, C. McDiarmid (1998) Minimum length spanning trees in regular graphs Combinatorica 18, 311–333.
R.M. Dudley (1989) Real Analysis and Probability Wadworth Publishing, Pacific Grove CA.
A.M. Frieze (1985) On the value of a random minimum spanning tree problem Discrete Appl. Math. 10, 47–56.
A.M. Frieze and C.J.H. McDiarmid (1989) On random minimum lenght spanning trees Combinatorica, 9, 363–374.
A. M. Frieze, M. Ruszinkó, and L. Thoma (2000) A note on random minimum lenght spanning trees Electronic Journal of Combinatorics 7, Research Paper 5, (5 pp.)
I.M. Gessel and B.E. Sagan (1996) The Tutte polynomial of a graph depth-first search and simpicial complex partitions Electronic Journal of Combinatorics 3, Reseach Paper 9, (36 pp.)
T.H. Harris (1989) The Theory of Branching Processes Dover Publications, New York.
S. Janson (1995) The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph Random Structures and Algorithms 7, 337–355.
D. Karger (1999) A randomized fully polynomial time approximation scheme for the all-terminal network reliablity problem SIAM Journal of Computing 29, 492–514.
J.F.C. Kingman (1993) Poisson Processes Oxford Universtiy Press, New York, 1993.
P.A.P. Moran (1967) A non-Markovian quasi-Poisson process Studia Scientiarum Mathematicarum Hungarica 2, 425–429.
S. Negami (1987) Polynomial Invariants of Graphs Transactions of the American Mathematical Society 299, 601–622.
G. Parisi (1998) A conjecture on radon’ bipartite matching ArXiv Condmat 980–1176.
M. Penrose (1998) Random minimum spanning tree and percolation on the n-cube Random Structures and Algorithms 12, 63–82.
A. Rényi, (1967) Remarks on the Poisson Process Studia Scientiarum Mathematicarum Hungarica 2, 119–123.
J.M. Steele(1987) On Frieze’s ζ(3)limit for lengths of minimal spanning trees Discrete Applied Mathematics, 18 (1987), 99–103.
J.M. Steele (1997) Probability Theory and Combinatorial Optimization NSFCBMS Volume 69. Society for Industrial and Applied Mathematics, Philadelphia.
D. Welsh (1999) The Tutte Polynomial Random Structures and Algorithms 15, 210–228.
D. Williams (1991) Probability with Martingales Cambridge University Press, Cambridge.
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Steele, J.M. (2002). Minimal Spanning Trees for Graphs with Random Edge Lengths. In: Chauvin, B., Flajolet, P., Gardy, D., Mokkadem, A. (eds) Mathematics and Computer Science II. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8211-8_14
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DOI: https://doi.org/10.1007/978-3-0348-8211-8_14
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